ECON402: Practice Final Exam Solutions

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1 CO42: Practice Final xam Solutions Summer 22 Instructions There is a total of four problems. You must answer any three of them. You get % for writing your name and 3% for each of the three best problems that you answer. You have minutes to complete the exam. Please don t forget to write your name on all the pages that you want to be graded. Bonus points Please write your name and report the grade that you expect to receive on the attached piece of paper. I will collect these papers and keep them in my office while the TA grades the exam. If your reported grade is close enough to the grade assigned by the TA, you will receive an additional % bonus. Formally, if you write the number X and the TA gives you a grade Y and X Y <2., then your final grade for the exam will be. Y Formulas Suppose that you want to find the value x that maximizes a concave and quadratic function f :R R. If f is written as f(x)= ax 2 + bx+ c with a>, then x = b/2a. If f is written as f(x)= a x x x x2 with a>, then x =(x + x 2 )/2. The present value of a constant stream of payoffs is ū/( δ). If v is distributed uniform[, v], then Prob(v v )= v / v. 3 pts Suppose that Anna and Bob play the following simultaneous move game twice, and Problem. their total payoffs are the sum of the payoffs they get at each stage. Is there a SP on which the players choose (B,y) on the first stage? If your answer is positive you must provide the SP describing the strategies in detail, if it is negative you must explain why this is the case. A, 2, 4, 3 B 2, 3,, 2 C, 4, 6 3, D 2, 2, 9, 2 There seems to be some confusion about this problem. A lot of people got the equivalent problem wrong on the HW so I will write a detailed description. This problem refers a to a simple extensive form game in which a strategic form game is played twice. The problem asks you if there is a SP in which players choose (B,y) on the first period. To find SP we use backward induction however: there is more than one way to do backward induction and there are many SP. I am only asking you to find one SP with the desired property.

2 Its important to start by understanding what is a strategy in this FG. A strategy has to specify what each player chooses on each information set. Both players have one information set on the first period and one information set on the second period corresponding to each different outcome of the first period. So each player has 2+=3 information sets. And a strategy must specify what to do on each of these information sets. A simple way of describing a strategy is to specify what players do on the first period (eg D,z) and to specify what they would do on the second period conditional on the outcome of the first period by using a table like the following one: A A,y D,y D,y B D,z C,x C,z C D,x B,z B,x D C,z A,x A,x This table specifies, for instance, that if the outcome of the first period is (A,x) then Anna will choose A on the second period and Bob will choose y on the second period. otice that there are many different ways of filling this table. Actually player has 4 3= and player 2 has 3 3=94323 different strategies, so that trying to write a complete strategic form game is pointless. SP requires that on the second period players choose a of the original game. Hence in the second period players have to choose either (A,x), (C,y) or (D,z) that would result in second period payoffs of (-,), (4,6) or (D,z) respectively. Depending on which you pick for each subgame you would different SP, I am asking you to pick them in such a way that you will find a SP in which they play (B,y) on the first period. For example you could specify the following strategy for the second period: A A,x D,z D,z B D,z A,x C,y C C,y A,x C,y D D,z C,y A,x Cont. strategies A -, 9,2 9,2 B 9,2 -, 4,6 C 4,6 -, 4,6 D 9,2 4,6 -, Cont. values A -2,,3 3, B 7, -,,8 C 3, 3, 7, D 7,3 6,6 8,7 Total. payoffs However choosing this strategies for the second period wont help our cause because they wont generate incentives to play (B,y) on the first period. By doing backward induction in this way you will find some SP but not the one that we are looking for. To see this notice that after replacing the subgames with the equilibrium payoffs we get the previous total payoff table (obtained by adding up the (second stage) continuation payoffs and the (first stage) payoffs from the original game). Which actually has no in pure strategies. To find the desired SP we must choose the continuation values (payoffs corresponding to the second period) in a smart way that generates the incentives too play (B,y) on the first period. We want to use continuation strategies with the following property: if people actually play (B,y) play a continuation equilibrium tomorrow that is good for both (eg (C,y)) if only Anna deviates play a continuation equilibrium that is bad for Anna (eg (A,x)) if only Bob deviates play a continuation equilibrium that is bad for Bob (eg (D,z)) 2

3 Consider the strategies with Anna and Bob choosing (B,y) on the first period and making choices in the second period that depend on the outcome of the first period according to the following table of continuation strategies: A C,y A,x C,y B D,z C,y D,z C C,y A,x C,y D C,y A,x C,y Cont. strategies A 4,6 -, 4,6 B 9,2 4,6 9,2 C 4,6 -, 4,6 D 4,6 -, 4,6 Cont. values A 3,,6 8,9 B 7, 4,6,4 C 3, 3, 7, D 2,7, 3,8 Total. payoffs Since (C,y), (D,z) and (A,x) are of the stage game, these strategies induce in all the proper subgames (in the second period). Performing one step of backward induction, we obtain the previous table of total payoffs for the first period. Since (B,y) is a of the resulting game, the proposed strategies are indeed a SP. 3 pts Consider a Cournot duopoly with firms and 2 producing the same good with constant Problem 2. marginal cost c= and inverse demand function: ach firm wishes to maximize its profits: P(q,q 2 )= q q 2 Best response functions are given by: u i (q,q 2 )=(P(q,q 2 ) c)q i =(9 q q 2 )q i BR i (q i )=4 2 q i The symmetric Pareto efficient outcome has both firms choosing q = 22 and the unique ash equilibrium has both firms producing q C = 3. ow suppose that the game is repeated infinitely and firms discount their future profits with a common discount factorδ (,). Consider the following grim trigger strategies: Choose q as long as everyone has chosen q in the past and choose p C otherwise Find the minimum value ofδsuch that the grim trigger strategies are a SP of the repeated game? Hint: Start by looking for the the most profitable deviations along the equilibrium path (ie the best responses to q in the stage game) Since(q C,q C ) is a of the stage game, it is always IC to play q C forever if you think that your opponent will play q C forever independently of what you do. Hence we only have to verify that players are willing to choose q along the equilibrium path. First notice that your most profitable deviation if your opponent is choosing q is q : q = BR(q )=4 2 22=337. 3

4 ow define u,u C and u to be: u = u(q C,q C )=( ) 22=2 u C = u(q C,q C )=(9 3 3) 3=9 u = u(q,q )=( ) 337.=396.2 So each player thinks as follows: If I choose q today I will get u forever whereas if I deviate today I can get at most u today and then I will get u C from tomorrow onwards. Hence a player wants to deviate if and only if: δ u < u + δ δ uc δ.3 For details as to how I got that value see slides on repeated games (we solved this problem in class). 3 pts Consider the following strategic situation with incomplete information involving two Problem 3. firms: an incumbent and a potential entrant. The incumbent is the only firm currently operating in the industry and it is sells a product with either good quality (G) or bad quality (B). The quality of the incumbent s product is known by only the incumbent, it is not known by the entrant. The product has good quality (G) with probability /2 and a bad quality (B) with probability /2. The game begins with the incumbent choosing between a high price (H) and low price (L). After observing this price, the entrant decides whether to enter () or not to enter (). Payoffs are as follows: H 4, 8, L, 6, G(/2) H, 3, L,, B(/2) 3.a) Write down an extensive form game with ature that represents this situation. The game should begin with ature choosing the type (cost) of the incumbent. L I H 4 6 G(/2) 8 B(/2) L I H 3 4

5 3.b) Write down a Bayesian strategic form game that represents this situation. HH., 2., 2.,., HL., 2.,. 3.,. 4., LH 2, 2., 2. 4, 2. 4., LL 2, 2 3., 2, 2 3., 3.c) Find all the Bayesian ash equilibria in pure strategies. The only B in pure strategies is (LL, ), ie the incumbent always chooses low prices and the incumbent chooses to enter if and only if it observes a high price. 3 pts Consider a first price auction with 2 potential buyers. ach buyer i has a private value Problem 4. v i and values are distributed uniformly on[,]. Buyers simultaneously and independently submit bids b i. The object is allocated to the buyer with the greatest bid so that payoffs are: v i b i if b i > b i u i (v, b)= otherwise Find the unique ash equilibrium in pure strategies. Hint: guess that equilibrium bids are linear, ie b i (v i )=av i See class slides or chapter 27 in the textbook for the solution.